mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.LocalProperties.Semilocal

Goal state before:
pp: case h
R : Type u_1
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Finite (MaximalSpectrum R)
M : Type u_2
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
Rₚ : (P : Ideal R) → [P.IsMaximal] → Type u_3
inst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)
inst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)
inst✝⁵ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalization.AtPrime (Rₚ P) P
Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4
inst✝⁴ : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)
inst✝³ : (P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)
inst✝² : (P : Ideal R) → [inst : P.IsMaximal] → Module (Rₚ P) (Mₚ P)
inst✝¹ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsScalarTower R (Rₚ P) (Mₚ P)
f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P
inst✝ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)
H : ∀ (P : Ideal R) [inst : P.IsMaximal], Module.Finite (Rₚ P) (Mₚ P)
this : Fintype (MaximalSpectrum R)
s : (P : MaximalSpectrum R) → Finset (Mₚ P.asIdeal)
hs : ∀ (P : MaximalSpectrum R), Submodule.span (Rₚ P.asIdeal) ↑(s P) = ⊤
frac : (P : MaximalSpectrum R) → Mₚ P.asIdeal → M × ↥P.asIdeal.primeCompl
hfrac : ∀ (P : MaximalSpectrum R) (y : Mₚ P.asIdeal), (frac P y).2 • y = (f P.asIdeal) (frac P y).1
P : Ideal R
hP : P.IsMaximal
⊢ ↑(s { asIdeal := P, isMaximal := hP }) ⊆
    ↑(Submodule.span (Rₚ P) (⇑(f P) '' ↑(Finset.univ.biUnion fun P ↦ Finset.image (fun x ↦ (frac P x).1) (s P))))

Tactic:
intro x

Goal state after:
case h
R : Type u_1
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Finite (MaximalSpectrum R)
M : Type u_2
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
Rₚ : (P : Ideal R) → [P.IsMaximal] → Type u_3
inst✝⁷ : (P : Ideal R) → [inst : P.IsMaximal] → CommSemiring (Rₚ P)
inst✝⁶ : (P : Ideal R) → [inst : P.IsMaximal] → Algebra R (Rₚ P)
inst✝⁵ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalization.AtPrime (Rₚ P) P
Mₚ : (P : Ideal R) → [P.IsMaximal] → Type u_4
inst✝⁴ : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)
inst✝³ : (P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)
inst✝² : (P : Ideal R) → [inst : P.IsMaximal] → Module (Rₚ P) (Mₚ P)
inst✝¹ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsScalarTower R (Rₚ P) (Mₚ P)
f : (P : Ideal R) → [inst : P.IsMaximal] → M →ₗ[R] Mₚ P
inst✝ : ∀ (P : Ideal R) [inst : P.IsMaximal], IsLocalizedModule P.primeCompl (f P)
H : ∀ (P : Ideal R) [inst : P.IsMaximal], Module.Finite (Rₚ P) (Mₚ P)
this : Fintype (MaximalSpectrum R)
s : (P : MaximalSpectrum R) → Finset (Mₚ P.asIdeal)
hs : ∀ (P : MaximalSpectrum R), Submodule.span (Rₚ P.asIdeal) ↑(s P) = ⊤
frac : (P : MaximalSpectrum R) → Mₚ P.asIdeal → M × ↥P.asIdeal.primeCompl
hfrac : ∀ (P : MaximalSpectrum R) (y : Mₚ P.asIdeal), (frac P y).2 • y = (f P.asIdeal) (frac P y).1
P : Ideal R
hP : P.IsMaximal
x : Mₚ { asIdeal := P, isMaximal := hP }.asIdeal
⊢ x ∈ ↑(s { asIdeal := P, isMaximal := hP }) →
    x ∈ ↑(Submodule.span (Rₚ P) (⇑(f P) '' ↑(Finset.univ.biUnion fun P ↦ Finset.image (fun x ↦ (frac P x).1) (s P))))
Module: Mathlib.RingTheory.LocalProperties.ProjectiveDimension

Goal state before:
pp: case succ
R : Type u
inst✝³ : CommRing R
inst✝² : Small.{v, u} R
inst✝¹ : IsNoetherianRing R
n : ℕ
M : ModuleCat R
inst✝ : Module.Finite R ↑M
P : Type v
w✝³ : AddCommGroup P
w✝² : Module R P
w✝¹ : Module.Free R P
w✝ : Module.Finite R P
f : P →ₗ[R] ↑M
ih :
  HasProjectiveDimensionLT (of R ↥f.ker) (n + 1) ↔
    ∀ (m : MaximalSpectrum R), HasProjectiveDimensionLT ((of R ↥f.ker).localizedModule m.asIdeal.primeCompl) (n + 1)
surjf : Function.Surjective ⇑f
S : ShortComplex (ModuleCat R) := f.shortComplexKer
S_exact : f.shortComplexKer.ShortExact
proj : Projective S.X₂
Sp : (p : MaximalSpectrum R) → ShortComplex (ModuleCat (Localization p.asIdeal.primeCompl)) :=
  fun p ↦ S.map (localizedModuleFunctor p.asIdeal.primeCompl)
Sp_exact : ∀ (p : MaximalSpectrum R), (Sp p).ShortExact
projp : ∀ (p : MaximalSpectrum R), Projective (Sp p).X₂
⊢ (∀ (m : MaximalSpectrum R), HasProjectiveDimensionLT ((of R ↥f.ker).localizedModule m.asIdeal.primeCompl) (n + 1)) ↔
    ∀ (m : MaximalSpectrum R), HasProjectiveDimensionLT (M.localizedModule m.asIdeal.primeCompl) (n + 1 + 1)

Tactic:
exact (forall_congr' (fun p ↦ (Sp_exact p).hasProjectiveDimensionLT_X₃_iff n (projp p))).symm

Goal state after:
Module: Mathlib.RingTheory.LocalProperties.ProjectiveDimension

Goal state before:
pp: case bot
R : Type u
inst✝³ : CommRing R
inst✝² : Small.{v, u} R
inst✝¹ : IsNoetherianRing R
M : ModuleCat R
inst✝ : Module.Finite R ↑M
aux : ∀ (n : ℕ), projectiveDimension M ≤ ↑n ↔ ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ↑n
⊢ projectiveDimension M ≤ ⊥ ↔ ⨆ p, projectiveDimension (M.localizedModule p.asIdeal.primeCompl) ≤ ⊥

Tactic:
simp only [le_bot_iff, projectiveDimension_eq_bot_iff, ModuleCat.isZero_iff_subsingleton,
      iSup_eq_bot, ModuleCat.localizedModule, ← Equiv.subsingleton_congr (equivShrink _)]
    refine ⟨fun h p ↦ LocalizedModule.instSubsingleton _, fun h ↦ ?_⟩
    apply Module.subsingleton_of_localization_maximal (R := R)
      (fun p ↦ LocalizedModule p.primeCompl M) (fun p ↦ LocalizedModule.mkLinearMap p.primeCompl M)
    intro p hp
    exact h ⟨p, hp⟩

Goal state after:
Module: Mathlib.RingTheory.LocalRing.Etale

Goal state before:
pp: case mp.refine_1
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : FaithfulSMul R S
inst✝ : FormallyUnramified R S
β : S
hβ : Function.Surjective ⇑(aeval ((residue S) β))
s : S
a✝ : s ∈ ⊤
q : R[X]
hp : (aeval ((residue S) β)) (map (residue R) q) = (residue S) s
⊢ (aeval β) q ∈ Subalgebra.toSubmodule R[β]

Tactic:
rw [adjoin_singleton_eq_range_aeval]

Goal state after:
case mp.refine_1
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : FaithfulSMul R S
inst✝ : FormallyUnramified R S
β : S
hβ : Function.Surjective ⇑(aeval ((residue S) β))
s : S
a✝ : s ∈ ⊤
q : R[X]
hp : (aeval ((residue S) β)) (map (residue R) q) = (residue S) s
⊢ (aeval β) q ∈ Subalgebra.toSubmodule (aeval β).range
Module: Mathlib.RingTheory.LocalRing.Etale

Goal state before:
pp: case mpr
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : FaithfulSMul R S
inst✝ : FormallyUnramified R S
β : S
hβ_gen : Function.Surjective ⇑(aeval β)
⊢ Function.Surjective ⇑(aeval ((residue S) β))

Tactic:
intro x

Goal state after:
case mpr
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : FaithfulSMul R S
inst✝ : FormallyUnramified R S
β : S
hβ_gen : Function.Surjective ⇑(aeval β)
x : ResidueField S
⊢ ∃ a, (aeval ((residue S) β)) a = x
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
x : R⸨X⸩
⊢ (ofPowerSeries ℤ R) x.powerSeriesPart = (single (-order x)) 1 * x

Tactic:
refine Eq.trans ?_ (congr rfl x.single_order_mul_powerSeriesPart)

Goal state after:
R : Type u_1
inst✝ : Semiring R
x : R⸨X⸩
⊢ (ofPowerSeries ℤ R) x.powerSeriesPart =
    (single (-order x)) 1 * ((single (order x)) 1 * (ofPowerSeries ℤ R) x.powerSeriesPart)
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: case neg
K : Type u_2
inst✝ : Field K
P : K[X]
hP : ¬P = 0
⊢ exp (-↑((Associates.mk (Polynomial.idealX K).asIdeal).count (Associates.mk (Ideal.span {P})).factors)) =
    (idealX K).intValuation ↑P

Tactic:
intValuation_if_neg _ <| (by simp [hP])

Goal state after:
case neg
K : Type u_2
inst✝ : Field K
P : K[X]
hP : ¬P = 0
⊢ exp (-↑((Associates.mk (Polynomial.idealX K).asIdeal).count (Associates.mk (Ideal.span {P})).factors)) =
    exp (-↑((Associates.mk (idealX K).asIdeal).count (Associates.mk (Ideal.span {↑P})).factors))
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: K : Type u_2
inst✝ : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
⊢ ∃ N, ∀ᶠ (f : K⸨X⸩) in ℱ, ∀ n < N, f.coeff n = 0

Tactic:
by
  let entourage : Set (K⸨X⸩ × K⸨X⸩) := {P : K⸨X⸩ × K⸨X⸩ | Valued.v.restrict (P.snd - P.fst) < 1}
  let ζ : (MonoidWithZeroHom.ValueGroup₀ <| .ofClass (Valued.v (R := K⸨X⸩)))ˣ :=
    Units.mk0 1 (zero_ne_one.symm)
  obtain ⟨S, ⟨hS, ⟨T, ⟨hT, H⟩⟩⟩⟩ := mem_prod_iff.mp <| Filter.le_def.mp hℱ.2 entourage
    <| (Valued.hasBasis_uniformity K⸨X⸩ ℤᵐ⁰).mem_of_mem (i := ζ) (by tauto)
  obtain ⟨f, hf⟩ := forall_mem_nonempty_iff_neBot.mpr hℱ.1 (S ∩ T) (inter_mem_iff.mpr ⟨hS, hT⟩)
  obtain ⟨N, hN⟩ : ∃ N : ℤ, ∀ g : K⸨X⸩,
    Valued.v (g - f) ≤ 1 → ∀ n < N, g.coeff n = 0 := by
    by_cases hf : f = 0
    · refine ⟨0, fun x hg ↦ ?_⟩
      rw [hf, sub_zero] at hg
      exact (valuation_le_iff_coeff_lt_eq_zero K).mp hg
    · refine ⟨min (f.2.isWF.min (HahnSeries.support_nonempty_iff.mpr hf)) 0 - 1, fun _ hg n hn ↦ ?_⟩
      rw [eq_coeff_of_valuation_sub_lt K hg (d := 0)]
      · exact Function.notMem_support.mp fun h ↦
        f.2.isWF.not_lt_min (HahnSeries.support_nonempty_iff.mpr hf) h
        <| lt_trans hn <| Int.sub_one_lt_iff.mpr <| min_le_left _ _
      exact lt_of_lt_of_le hn <| le_of_lt (Int.sub_one_lt_of_le <| min_le_right _ _)
  use N
  apply mem_of_superset (inter_mem hS hT)
  intro g hg
  have h_prod : (f, g) ∈ S ×ˢ T := by simp [hf.1, hg.2]
  refine hN g (le_of_lt ?_)
  simpa [Valuation.restrict_def, ← Valuation.restrict_lt_one_iff] using! H h_prod

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: case refine_3
R : Type u_3
m : ℕ
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
i : ℕ
him : i < m
this :
  supDegree (⇑toLex) (esymm (Fin m) R (i + 1)) =
      supDegree (⇑toLex) (single (Finsupp.indicator (Iic ⟨i, him⟩) fun x x_1 ↦ 1) 1) ∧
    leadingCoeff (⇑toLex) (esymm (Fin m) R (i + 1)) =
      leadingCoeff (⇑toLex) (single (Finsupp.indicator (Iic ⟨i, him⟩) fun x x_1 ↦ 1) 1)
⊢ supDegree (⇑toLex) (esymm (Fin m) R (i + 1)) = toLex (Finsupp.indicator (Iic ⟨i, him⟩) fun x x_1 ↦ 1) ∧
    Monic (⇑toLex) (esymm (Fin m) R (i + 1))

Tactic:
supDegree_single_ne_zero _ one_ne_zero,

Goal state after:
case refine_3
R : Type u_3
m : ℕ
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
i : ℕ
him : i < m
this :
  supDegree (⇑toLex) (esymm (Fin m) R (i + 1)) = toLex (Finsupp.indicator (Iic ⟨i, him⟩) fun x x_1 ↦ 1) ∧
    leadingCoeff (⇑toLex) (esymm (Fin m) R (i + 1)) =
      leadingCoeff (⇑toLex) (single (Finsupp.indicator (Iic ⟨i, him⟩) fun x x_1 ↦ 1) 1)
⊢ supDegree (⇑toLex) (esymm (Fin m) R (i + 1)) = toLex (Finsupp.indicator (Iic ⟨i, him⟩) fun x x_1 ↦ 1) ∧
    Monic (⇑toLex) (esymm (Fin m) R (i + 1))
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: case h
K : Type u_2
inst✝ : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : Multiplicative.toAdd (unzero ⋯) ≤ 0
d : ℕ
hd : Multiplicative.toAdd (unzero ⋯) = -↑d
this : Valued.v ((algebraMap K⟦X⟧ K⸨X⸩) (F - ↑((trunc (d + 1)) F))) ≤ ↑η * ↑(Multiplicative.ofAdd (-1))
⊢ (PowerSeries.idealX K).intValuation (F - ↑((trunc (d + 1)) F)) < ↑η

Tactic:
rw [← valuation_of_algebraMap (K := K⸨X⸩) (PowerSeries.idealX K) (F - F.trunc (d + 1))]

Goal state after:
case h
K : Type u_2
inst✝ : Field K
F : K⟦X⟧
η : (WithZero (Multiplicative ℤ))ˣ
h_neg : Multiplicative.toAdd (unzero ⋯) ≤ 0
d : ℕ
hd : Multiplicative.toAdd (unzero ⋯) = -↑d
this : Valued.v ((algebraMap K⟦X⟧ K⸨X⸩) (F - ↑((trunc (d + 1)) F))) ≤ ↑η * ↑(Multiplicative.ofAdd (-1))
⊢ (valuation K⸨X⸩ (PowerSeries.idealX K)) ↑(F - ↑((trunc (d + 1)) F)) < ↑η
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: case refine_2
R : Type u_3
m : ℕ
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
i : ℕ
him : i < m
t : Finset (Fin m)
ht : t ∈ powersetCard (i + 1) univ
hne : t ≠ Iic ⟨i, him⟩
ht' : #t = #(Iic ⟨i, him⟩)
⊢ supDegree (⇑toLex) (single (∑ j ∈ t, fun₀ | j => 1) 1) < supDegree (⇑toLex) (single (∑ j ≤ ⟨i, him⟩, fun₀ | j => 1) 1)

Tactic:
supDegree_single_ne_zero _ one_ne_zero,

Goal state after:
case refine_2
R : Type u_3
m : ℕ
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
i : ℕ
him : i < m
t : Finset (Fin m)
ht : t ∈ powersetCard (i + 1) univ
hne : t ≠ Iic ⟨i, him⟩
ht' : #t = #(Iic ⟨i, him⟩)
⊢ toLex (∑ j ∈ t, fun₀ | j => 1) < toLex (∑ j ≤ ⟨i, him⟩, fun₀ | j => 1)
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝¹ : CommSemiring R
inst✝ : LinearOrder σ
p : MvPolynomial σ R
hp : p.IsSymmetric
⊢ Antitone ⇑(ofLex (supDegree (⇑toLex) p))

Tactic:
obtain rfl | h0 := eq_or_ne p 0

Goal state after:
case inl
σ : Type u_1
R : Type u_3
inst✝¹ : CommSemiring R
inst✝ : LinearOrder σ
hp : IsSymmetric 0
⊢ Antitone ⇑(ofLex (supDegree (⇑toLex) 0))

case inr
σ : Type u_1
R : Type u_3
inst✝¹ : CommSemiring R
inst✝ : LinearOrder σ
p : MvPolynomial σ R
hp : p.IsSymmetric
h0 : p ≠ 0
⊢ Antitone ⇑(ofLex (supDegree (⇑toLex) p))
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: case right
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
⊢ ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range

Tactic:
obtain rfl | h0 := eq_or_ne p 0

Goal state after:
case right.inl
R : Type u_3
inst✝ : CommRing R
n : ℕ
hp : 0 ∈ symmetricSubalgebra (Fin n) R
⊢ ⟨0, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range

case right.inr
R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
⊢ ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem

Goal state before:
pp: R : Type u_3
inst✝ : CommRing R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ∈ symmetricSubalgebra (Fin n) R
h0 : p ≠ 0
ih :
  ∀ y < supDegree (⇑toLex) p,
    ∀ (p : MvPolynomial (Fin n) R) (hp : p ∈ symmetricSubalgebra (Fin n) R),
      p ≠ 0 → supDegree (⇑toLex) p = y → ⟨p, hp⟩ ∈ (esymmAlgHom (Fin n) R n).range
t : Fin n →₀ ℕ := Finsupp.equivFunOnFinite.symm (invAccumulate n n ⇑(ofLex (supDegree (⇑toLex) p)))
⊢ (accumulate n n) ⇑t = ⇑(ofLex (supDegree (⇑toLex) p))

Tactic:
exact accumulate_invAccumulate le_rfl hp.antitone_supDegree

Goal state after:
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: case hf
K : Type u_2
inst✝ : Field K
⊢ ∀ (x : WithVal (polynomialValuationX K)), Valued.v ((LaurentSeriesPkg K).coe x) = Valued.v x

Tactic:
intro x

Goal state after:
case hf
K : Type u_2
inst✝ : Field K
x : WithVal (polynomialValuationX K)
⊢ Valued.v ((LaurentSeriesPkg K).coe x) = Valued.v x
Module: Mathlib.RingTheory.MvPowerSeries.Restricted

Goal state before:
pp: M : Type u_3
S : Type u_4
inst✝³ : AddMonoid M
inst✝² : Finset.HasAntidiagonal M
inst✝¹ : NormedRing S
inst✝ : IsUltrametricDist S
f g : M → S
C : M → ℝ
hC : ∀ (a b : M), C (a + b) = C a * C b
hf : Tendsto (fun i ↦ ‖f i‖ * C i) cofinite (𝓝 0)
hg : Tendsto (fun i ↦ ‖g i‖ * C i) cofinite (𝓝 0)
hC' : 0 ≤ C
a : M
this : 0 ≤ C a
x : M × M
hx : x ∈ antidiagonal a
⊢ ‖f x.1 * g x.2‖ * C a ≤ ‖f x.1‖ * C x.1 * (‖g x.2‖ * C x.2)

Tactic:
mul_mul_mul_comm,

Goal state after:
M : Type u_3
S : Type u_4
inst✝³ : AddMonoid M
inst✝² : Finset.HasAntidiagonal M
inst✝¹ : NormedRing S
inst✝ : IsUltrametricDist S
f g : M → S
C : M → ℝ
hC : ∀ (a b : M), C (a + b) = C a * C b
hf : Tendsto (fun i ↦ ‖f i‖ * C i) cofinite (𝓝 0)
hg : Tendsto (fun i ↦ ‖g i‖ * C i) cofinite (𝓝 0)
hC' : 0 ≤ C
a : M
this : 0 ≤ C a
x : M × M
hx : x ∈ antidiagonal a
⊢ ‖f x.1 * g x.2‖ * C a ≤ ‖f x.1‖ * ‖g x.2‖ * (C x.1 * C x.2)
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: case cont_f
K : Type u_2
inst✝ : Field K
f : K⸨X⸩
this : UniformSpace ratfuncAdicComplPkg.space := ratfuncAdicComplPkg.uniformStruct
raw_surj : Function.Surjective ⇑Valued.v
⊢ Continuous ⇑Valued.v

pp: case a
K : Type u_2
inst✝ : Field K
f : K⸨X⸩
this : UniformSpace ratfuncAdicComplPkg.space := ratfuncAdicComplPkg.uniformStruct
raw_surj : Function.Surjective ⇑Valued.v
⊢ ∀ (a : ratfuncAdicComplPkg.space),
    Tendsto (⇑Valued.v) (Filter.comap ratfuncAdicComplPkg.coe (𝓝 a)) (𝓝 (⋯.extend (⇑Valued.v) a))

Tactic:
· refine Valued.continuous_valuation_of_surjective (fun x ↦ ?_)
    obtain ⟨y, rfl⟩ := RatFunc.valuation_surjective K x
    exact ⟨.toVal _ y, rfl⟩

Goal state after:
case a
K : Type u_2
inst✝ : Field K
f : K⸨X⸩
this : UniformSpace ratfuncAdicComplPkg.space := ratfuncAdicComplPkg.uniformStruct
raw_surj : Function.Surjective ⇑Valued.v
⊢ ∀ (a : ratfuncAdicComplPkg.space),
    Tendsto (⇑Valued.v) (Filter.comap ratfuncAdicComplPkg.coe (𝓝 a)) (𝓝 (⋯.extend (⇑Valued.v) a))
Module: Mathlib.RingTheory.LaurentSeries

Goal state before:
pp: case a
K : Type u_2
inst✝ : Field K
f : K⸨X⸩
this : UniformSpace ratfuncAdicComplPkg.space := ratfuncAdicComplPkg.uniformStruct
raw_surj : Function.Surjective ⇑Valued.v
⊢ ∀ (a : ratfuncAdicComplPkg.space),
    Tendsto (⇑Valued.v) (Filter.comap ratfuncAdicComplPkg.coe (𝓝 a)) (𝓝 (⋯.extend (⇑Valued.v) a))

Tactic:
intro x

Goal state after:
case a
K : Type u_2
inst✝ : Field K
f : K⸨X⸩
this : UniformSpace ratfuncAdicComplPkg.space := ratfuncAdicComplPkg.uniformStruct
raw_surj : Function.Surjective ⇑Valued.v
x : ratfuncAdicComplPkg.space
⊢ Tendsto (⇑Valued.v) (Filter.comap ratfuncAdicComplPkg.coe (𝓝 x)) (𝓝 (⋯.extend (⇑Valued.v) x))
Module: Mathlib.RingTheory.NoetherNormalization

Goal state before:
pp: k : Type u_1
inst✝ : Field k
n : ℕ
f : MvPolynomial (Fin (n + 1)) k
fne : f ≠ 0
v : Fin (n + 1) →₀ ℕ
vin : v ∈ f.support
h : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)
vs :
  ∀ x' ∈ f.support,
    ((MvPolynomial.finSuccEquiv k n) ((T f) ((MvPolynomial.monomial x') (MvPolynomial.coeff x' f)))).natDegree ≤
      ((MvPolynomial.finSuccEquiv k n) ((T f) ((MvPolynomial.monomial v) (MvPolynomial.coeff v f)))).natDegree
⊢ ∀ x ∈ f.support \ {v},
    ((MvPolynomial.finSuccEquiv k n) ((T f) (h x))).degree < ((MvPolynomial.finSuccEquiv k n) ((T f) (h v))).degree

Tactic:
intro x

Goal state after:
k : Type u_1
inst✝ : Field k
n : ℕ
f : MvPolynomial (Fin (n + 1)) k
fne : f ≠ 0
v : Fin (n + 1) →₀ ℕ
vin : v ∈ f.support
h : (Fin (n + 1) →₀ ℕ) → MvPolynomial (Fin (n + 1)) k := fun w ↦ (MvPolynomial.monomial w) (MvPolynomial.coeff w f)
vs :
  ∀ x' ∈ f.support,
    ((MvPolynomial.finSuccEquiv k n) ((T f) ((MvPolynomial.monomial x') (MvPolynomial.coeff x' f)))).natDegree ≤
      ((MvPolynomial.finSuccEquiv k n) ((T f) ((MvPolynomial.monomial v) (MvPolynomial.coeff v f)))).natDegree
x : Fin (n + 1) →₀ ℕ
⊢ x ∈ f.support \ {v} →
    ((MvPolynomial.finSuccEquiv k n) ((T f) (h x))).degree < ((MvPolynomial.finSuccEquiv k n) ((T f) (h v))).degree
Module: Mathlib.RingTheory.WittVector.Basic

Goal state before:
pp: p : ℕ
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
α : Type u_3
β : Type u_4
inst✝ : Invertible ↑p
⊢ LeftInverse (fun x ↦ mk p fun n ↦ (aeval x) (xInTermsOfW p R n)) ghostFun

Tactic:
intro x

Goal state after:
p : ℕ
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
α : Type u_3
β : Type u_4
inst✝ : Invertible ↑p
x : 𝕎 R
⊢ (fun x ↦ mk p fun n ↦ (aeval x) (xInTermsOfW p R n)) x.ghostFun = x
Module: Mathlib.RingTheory.WittVector.Basic

Goal state before:
pp: p : ℕ
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
α : Type u_3
β : Type u_4
inst✝ : Invertible ↑p
⊢ RightInverse (fun x ↦ mk p fun n ↦ (aeval x) (xInTermsOfW p R n)) ghostFun

Tactic:
intro x

Goal state after:
p : ℕ
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
α : Type u_3
β : Type u_4
inst✝ : Invertible ↑p
x : ℕ → R
⊢ ((fun x ↦ mk p fun n ↦ (aeval x) (xInTermsOfW p R n)) x).ghostFun = x
Module: Mathlib.RingTheory.WittVector.IsPoly

Goal state before:
pp: p : ℕ
idx : Type u_1
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial (idx × ℕ) ℤ
h : ∀ (n : ℕ), (bind₁ f) (wittPolynomial p ℤ n) = (bind₁ g) (wittPolynomial p ℤ n)
⊢ f = g

Tactic:
ext1 n

Goal state after:
p : ℕ
idx : Type u_1
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial (idx × ℕ) ℤ
h : ∀ (n : ℕ), (bind₁ f) (wittPolynomial p ℤ n) = (bind₁ g) (wittPolynomial p ℤ n)
n : ℕ
⊢ f n = g n
Module: Mathlib.RingTheory.WittVector.IsPoly

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial ℕ ℤ
h : ∀ (n : ℕ), (bind₁ f) (wittPolynomial p ℤ n) = (bind₁ g) (wittPolynomial p ℤ n)
⊢ f = g

Tactic:
ext1 n

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ℕ → MvPolynomial ℕ ℤ
h : ∀ (n : ℕ), (bind₁ f) (wittPolynomial p ℤ n) = (bind₁ g) (wittPolynomial p ℤ n)
n : ℕ
⊢ f n = g n
Module: Mathlib.RingTheory.WittVector.Frobenius

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ (bind₁ (frobeniusPolyRat p)) (wittPolynomial p ℚ n) = wittPolynomial p ℚ (n + 1)

Tactic:
delta frobeniusPolyRat
  rw [← bind₁_bind₁, bind₁_xInTermsOfW_wittPolynomial, bind₁_X_right, Function.comp_apply]

Goal state after:
Module: Mathlib.RingTheory.WittVector.IsPoly

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), (ghostComponent n) (f x) = (ghostComponent n) (g x)
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n ↦ (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n ↦ (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n k : ℕ
⊢ ∀ (x : ℕ → ℤ),
    (MvPolynomial.eval x) ((bind₁ φ) (wittPolynomial p ℤ k)) = (MvPolynomial.eval x) ((bind₁ ψ) (wittPolynomial p ℤ k))

Tactic:
intro x

Goal state after:
p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [CommRing R] → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ), (ghostComponent n) (f x) = (ghostComponent n) (g x)
φ : ℕ → MvPolynomial ℕ ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (f x).coeff = fun n ↦ (aeval x.coeff) (φ n)
ψ : ℕ → MvPolynomial ℕ ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x : 𝕎 R), (g x).coeff = fun n ↦ (aeval x.coeff) (ψ n)
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ : 𝕎 R✝
n k : ℕ
x : ℕ → ℤ
⊢ (MvPolynomial.eval x) ((bind₁ φ) (wittPolynomial p ℤ k)) = (MvPolynomial.eval x) ((bind₁ ψ) (wittPolynomial p ℤ k))
Module: Mathlib.RingTheory.WittVector.StructurePolynomial

Goal state before:
pp: p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
inst✝ : Fintype idx
Φ : MvPolynomial idx ℚ
n : ℕ
⊢ ((bind₁ fun k ↦ (bind₁ fun i ↦ (rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n)).vars ⊆
    Finset.univ ×ˢ Finset.range (n + 1)

Tactic:
intro x

Goal state after:
p : ℕ
idx : Type u_2
hp : Fact (Nat.Prime p)
inst✝ : Fintype idx
Φ : MvPolynomial idx ℚ
n : ℕ
x : idx × ℕ
⊢ x ∈ ((bind₁ fun k ↦ (bind₁ fun i ↦ (rename (Prod.mk i)) (W_ ℚ k)) Φ) (xInTermsOfW p ℚ n)).vars →
    x ∈ Finset.univ ×ˢ Finset.range (n + 1)
Module: Mathlib.RingTheory.Perfection

Goal state before:
pp: R : Type u_1
inst✝¹ : CommSemiring R
p : ℕ
hp : Fact (Nat.Prime p)
inst✝ : CharP R p
h : Function.Surjective ⇑(frobenius R p)
n : ℕ
⊢ Function.Surjective ⇑(coeff R p n)

Tactic:
intro x

Goal state after:
R : Type u_1
inst✝¹ : CommSemiring R
p : ℕ
hp : Fact (Nat.Prime p)
inst✝ : CharP R p
h : Function.Surjective ⇑(frobenius R p)
n : ℕ
x : R
⊢ ∃ a, (coeff R p n) a = x
Module: Mathlib.RingTheory.WittVector.Verschiebung

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ ∀ (x : ℕ → ℤ),
    (MvPolynomial.eval x) ((bind₁ verschiebungPoly) (wittPolynomial p ℤ n)) =
      (MvPolynomial.eval x) (if n = 0 then 0 else ↑p * wittPolynomial p ℤ (n - 1))

Tactic:
intro x

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
x : ℕ → ℤ
⊢ (MvPolynomial.eval x) ((bind₁ verschiebungPoly) (wittPolynomial p ℤ n)) =
    (MvPolynomial.eval x) (if n = 0 then 0 else ↑p * wittPolynomial p ℤ (n - 1))
Module: Mathlib.RingTheory.WittVector.Identities

Goal state before:
pp: p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
this : IsPoly p fun {R} [CommRing R] x ↦ frobenius (verschiebung x)
⊢ frobenius (verschiebung x) = x * ↑p

Tactic:
have : IsPoly p fun {R} [CommRing R] x ↦ x * p := mulN_isPoly p p

Goal state after:
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x : 𝕎 R
this✝ : IsPoly p fun {R} [CommRing R] x ↦ frobenius (verschiebung x)
this : IsPoly p fun {R} [CommRing R] x ↦ x * ↑p
⊢ frobenius (verschiebung x) = x * ↑p